Numerical Differentiation - Examples & Applications

Numerical Differentiation - Examples & Applications

This section provides practical examples of numerical differentiation techniques, including forward, backward, and centered finite differences, high-accuracy formulas, Richardson extrapolation, unequally spaced data, and partial derivatives.

First-Derivative Finite Differences

The most basic methods for estimating the first derivative are the forward, backward, and centered divided difference formulas, all derived from the Taylor series expansion.

Intermediate: Comparing Forward, Backward, and Centered Differences

Estimate the first derivative of $f(x) = -0.1x^4 - 0.15x^3 - 0.5x^2 - 0.25x + 1.2$ at $x = 0.5$ using a step size of $h = 0.25$ . Calculate the true percent relative error for the forward, backward, and centered difference approximations. The true value of the derivative is $f'(0.5) = -0.9125$ .

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Higher-Accuracy Differentiation Formulas

By retaining more terms in the Taylor series expansion, we can derive finite difference formulas with higher-order accuracy (e.g., $O(h^2)$ for forward/backward, and $O(h^4)$ for centered).

Advanced: High-Accuracy Centered Difference

Estimate the first derivative of $f(x) = -0.1x^4 - 0.15x^3 - 0.5x^2 - 0.25x + 1.2$ at $x = 0.5$ using the $O(h^4)$ centered difference formula. Use a step size of $h = 0.25$ . The true value is $-0.9125$ .

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Derivatives of Unequally Spaced Data

When data points are not equally spaced, standard finite difference formulas cannot be used. Instead, a Lagrange interpolating polynomial is fit through the points and then differentiated.

Intermediate: Unequally Spaced Data

Given three unequally spaced points: $x_0 = 1, f(x_0) = 0$ $x_1 = 4, f(x_1) = 1.386$ $x_2 = 6, f(x_2) = 1.792$ Estimate the derivative $f'(4)$ .

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Partial Derivatives

Partial derivatives are estimated by holding all other variables constant and applying standard finite difference formulas to the variable of interest.

Intermediate: Numerical Partial Derivative

Let $f(x, y) = x^2 y + 3xy^2$ . Estimate $\frac{\partial f}{\partial x}$ at $(1, 2)$ using a centered difference with $h_x = 0.1$ .

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Richardson Extrapolation

Richardson extrapolation uses two derivative estimates computed with different step sizes to obtain a third, more accurate estimate.

Advanced: Richardson Extrapolation

Use Richardson extrapolation to estimate the first derivative of $f(x) = \ln(x)$ at $x = 2$ . First, calculate centered difference estimates using $h_1 = 0.2$ and $h_2 = 0.1$ . Then, combine them using Richardson extrapolation to find an $O(h^4)$ estimate. True value is $f'(2) = 1/2 = 0.5$ .

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